1. Technical Field of the Invention
The present invention relates to the removal of image artifacts, in particular to the removal of artifacts from phase encoded images.
2. Description of Related Art
It has been known in many different fields to phase-encode image data. One such field is the recently developed wavefront coding (WFC) technique, developed to increase the depth of field of incoherent optical systems and described in E. Dowski and T. W. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859-1866 (1995), the disclosure of which is hereby incorporated by reference.
In this approach, pupil-plane masks are designed to alter, that is to code, the transmitted incoherent wavefront so that the point-spread function (PSF) is almost constant near the focal plane and is highly extended in comparison with the conventional Airy pattern. As a consequence the wavefront coded image is distorted and can be accurately restored with digital processing for a wide range of defocus values. By jointly optimizing the optical coding and digital decoding, it is possible to achieve tolerance to defocus which could not be attained by traditional imaging systems while maintaining their diffraction-limited resolution.
The phase encoding principle is illustrated in FIG. 1. An optical system 10 comprises lenses and/or other optical elements and a phase encoding means 12 which is near to or in the pupil plane that changes the phase of the radiation that is incident upon it. The phase encoding means 12 can take the form of a phase mask. Due to the phase mask, the optical system 10 produces a phase encoded image 14 of an object 16, which is detected by image sensing means 18. The phase encoded image 14 appears blurred when viewed. Processing means 20 then applies a reconstruction algorithm to remove the phase encoding to produce a restored image 22, which appears in focus, that is, sharp, when viewed. Because the variation in the point spread function is predetermined by the choice of mask, the reconstruction algorithm can be written to reverse the blurring effects of the phase encoding means 12.
Various methods have been used for the design of the phase mask, for both square and circular apertures. Early design of phase masks was carried out in the frequency domain by the use of the ambiguity function (AF). The AF combined with the stationary phase approximation indicates that the ideal phase mask for extending the depth of field must be anti-symmetric and have a linear separable cubic form:f(x,y)=α(x3+y3)
A cubic phase mask 24 of this type is illustrated in FIG. 2. The strength of the phase mask, α, sets the maximum wavefront deviation and yields the amount of defocus invariance in the decoded image.
In the last five years, pupil plane encoding has been extended to include more general phase functions; phase masks have been successfully designed in the spatial domain in which the point spread function (PSF), Strehl ratio and Fisher information metrics are solved to be invariant to defocus. A technique called Pupil Plane Engineering has been developed by Prasad et al, and is described in S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, J. van der Gracht, “Engineering the Pupil Phase to Improve Image Quality,” in Proceedings of the SPIE, Vol. 5108 Visual Information Processing XII, edited by Z. Rahman, R. Schowengrdt, and S. Reichenbach (SPIE, Wellingham, Wash., 2003), pp. 1-12, the disclosure of which is hereby incorporated by reference. This approach generalizes the shape of the mask to include higher polynomial orders and is optimized by means of Fisher information metrics. The derived PPE mask 26, see FIG. 2, has an anti-symmetric phase shape (like a petal) and is given by:θ(x,y)=β(x3+y3)+γ(x2y+xy2)
where |x|<1, |y|<1 are normalized co-ordinates and β and γ are real variables that control the optical path difference (OPD) or amount of coding introduced in the transmitted wavefront of the optical system. We will denote by α the OPD introduced by a phase mask. For the 2D cubic phase mask, the maximum the peak-to-valley OPD is given by 4α.
In addition, radially symmetric quartic and logarithmic phase masks, which can be manufactured using traditional techniques, also enable aberration mitigation. The performance attained by these kinds of phase mask cannot equal that of anti-symmetric masks, but are suitable under modest amounts of aberrations and can be used without digital signal processing.
Phase coding and other phase perturbation techniques can greatly increase the depth of field which is useful for a wide range of applications and environments. However, this advantage is achieved at the expense of noise amplification during the decoding process. This effect increases with the strength of phase mask that is applied. Since the restored image suffers from a reduced signal-to-noise ratio, a trade-off between the signal-to-noise ratio of the restored image and its depth of field is required. Therefore, there is a noise cost that is inherent to the wavefront coding itself.
Furthermore, the PSF of such optical systems is usually considered to be shape invariant with defocus near the focal plane, so that the restoration of objects that belong to a specific range of defocus can be performed with a single kernel. However, there are in fact significant phase variations and amplitude variations of the Optical Transfer Function with defocus which cannot be compensated for by restoration using a single kernel. These variations in the OTF result in artifacts being embedded in the restored images which degrade the quality of the restored image.
These defects are generally poorly described in the literature, as they are not important in lower quality image systems. However, it is known how to remove these artifacts with specialized nonlinear filters. If the phase encoding means is rectangularly separable, the operation of the filters is also rectangularly separable so that computational speed can be increased through parallel processing.
Also, an approximate expression of the OTF of an optical system including a cubic phase mask is given in G. Muyo and A. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Letters, 2005, 2715-2717, the disclosure of which is hereby incorporated by reference. This paper shows how the decomposition of the optical transfer function (OTF) of a wavefront coding system can be described as a generalized Cornu spiral (GCS), the geometry of which can be used to estimate an approximation of the value for the phase and magnitude of the optical transfer function (OTF), the maximum value of a defocus coefficient for which the OTF can be considered to be approximately constant, the magnitude of the amplitude modulation of the MTF within the region of invariance, and the magnitude of phase modulation introduced. These features present in the OTF's of wavefront coded systems are explained analytically and so the possibility of simple calibration was introduced, that is, estimating the discrepancy between the coding and decoding kernels.
However, none of these methods provide for removal of artifacts from a restored phase encoded image.